Fall 2006 CENG 707 1

Algorithm Analysis

Fall 2006 CENG 707 2

Algorithmic Performance There are two aspects of algorithmic performance:• Time

• Instructions take time.• How fast does the algorithm perform?• What affects its runtime? 

• Space• Data structures take space• What kind of data structures can be used?• How does choice of data structure affect the runtime?

We will focus on time: – How to estimate the time required for an algorithm– How to reduce the time required

Fall 2006 CENG 707 3

Analysis of Algorithms

• When we analyze algorithms, we should employ mathematical techniques that analyze algorithms independently of specific implementations, computers, or data.

• To analyze algorithms:– First, we start to count the number of significant

operations in a particular solution to assess its efficiency.

– Then, we will express the efficiency of algorithms using growth functions.

Fall 2006 CENG 707 4

The Execution Time of Algorithms

• Each operation in an algorithm (or a program) has a cost.

Each operation takes a certain CPU time.

count = count + 1; take a certain amount of time, but it is constant

A sequence of operations:

count = count + 1; Cost: c1

sum = sum + count; Cost: c2

Total Cost = c1 + c2

Fall 2006 CENG 707 5

The Execution Time of Algorithms (cont.)

Example: Simple If-Statement

Cost Times

if (n < 0) c1 1

absval = -n c2 1

else

absval = n; c3 1

Total Cost <= c1 + max(c2,c3)

Fall 2006 CENG 707 6

The Execution Time of Algorithms (cont.)

Example: Simple Loop

Cost Times

i = 1; c1 1

sum = 0; c2 1

while (i <= n) { c3 n+1

i = i + 1; c4 n

sum = sum + i; c5 n

}

Total Cost = c1 + c2 + (n+1)*c3 + n*c4 + n*c5

The time required for this algorithm is proportional to n

Fall 2006 CENG 707 7

The Execution Time of Algorithms (cont.)Example: Nested Loop

Cost Timesi=1; c1 1sum = 0; c2 1while (i <= n) { c3 n+1

j=1; c4 nwhile (j <= n) { c5 n*(n+1) sum = sum + i; c6 n*n j = j + 1; c7 n*n

} i = i +1; c8 n}

Total Cost = c1 + c2 + (n+1)*c3 + n*c4 + n*(n+1)*c5+n*n*c6+n*n*c7+n*c8 The time required for this algorithm is proportional to n2

Fall 2006 CENG 707 8

General Rules for Estimation

• Loops: The running time of a loop is at most the running time of the statements inside of that loop times the number of iterations.

• Nested Loops: Running time of a nested loop containing a statement in the inner most loop is the running time of statement multiplied by the product of the sized of all loops.

• Consecutive Statements: Just add the running times of those consecutive statements.

• If/Else: Never more than the running time of the test plus the larger of running times of S1 and S2.

Fall 2006 CENG 707 9

Algorithm Growth Rates• We measure an algorithm’s time requirement as a function of the

problem size.– Problem size depends on the application: e.g. number of elements in a list for a

sorting algorithm, the number disks for towers of hanoi.

• So, for instance, we say that (if the problem size is n)– Algorithm A requires 5*n2 time units to solve a problem of size n.– Algorithm B requires 7*n time units to solve a problem of size n.

• The most important thing to learn is how quickly the algorithm’s time requirement grows as a function of the problem size.– Algorithm A requires time proportional to n2.– Algorithm B requires time proportional to n.

• An algorithm’s proportional time requirement is known as growth rate.

• We can compare the efficiency of two algorithms by comparing their growth rates.

Fall 2006 CENG 707 10

Algorithm Growth Rates (cont.)

Time requirements as a function of the problem size n

Fall 2006 CENG 707 11

Common Growth Rates

Function Growth Rate Name

c Constant

log N Logarithmic

log2N Log-squared

N Linear

N log N

N2 Quadratic

N3 Cubic

2N Exponential

Fall 2006 CENG 707 12

Figure 6.1Running times for small inputs

Fall 2006 CENG 707 13

Figure 6.2Running times for moderate inputs

Fall 2006 CENG 707 14

Order-of-Magnitude Analysis and Big 0 Notation

• If Algorithm A requires time proportional to f(n), Algorithm A is said to be order f(n), and it is denoted as O(f(n)).

• The function f(n) is called the algorithm’s growth-rate function.

• Since the capital O is used in the notation, this notation is called the Big O notation.

• If Algorithm A requires time proportional to n2, it is O(n2).

• If Algorithm A requires time proportional to n, it is O(n).

Fall 2006 CENG 707 15

Definition of the Order of an Algorithm

Definition:

Algorithm A is order f(n) – denoted as O(f(n)) –

if constants k and n0 exist such that A requires

no more than k*f(n) time units to solve a problem

of size n n0.

• The requirement of n n0 in the definition of O(f(n)) formalizes the notion of sufficiently large problems.– In general, many values of k and n0 can satisfy this definition.

Fall 2006 CENG 707 16

Order of an Algorithm

• Suppose an algorithm requires n2–3*n+10 seconds to solve a problem size n. If constants k and n0 exist such that

k*n2 > n2–3*n+10 for all n n0 .

the algorithm is order n2 (In fact, k is 3 and n0 is 2)

3*n2 > n2–3*n+10 for all n 2 .

Thus, the algorithm requires no more than k*n2 time units for n n0 ,

So it is O(n2)

Fall 2006 CENG 707 17

A Comparison of Growth-Rate Functions

Fall 2006 CENG 707 18

A Comparison of Growth-Rate Functions (cont.)

Fall 2006 CENG 707 19

Growth-Rate Functions

O(1) Time requirement is constant, and it is independent of the problem’s size.

O(log2n) Time requirement for a logarithmic algorithm increases increases slowly

as the problem size increases.

O(n) Time requirement for a linear algorithm increases directly with the size

of the problem.

O(n*log2n) Time requirement for a n*log2n algorithm increases more rapidly than

a linear algorithm.

O(n2) Time requirement for a quadratic algorithm increases rapidly with the

size of the problem.

O(n3) Time requirement for a cubic algorithm increases more rapidly with the

size of the problem than the time requirement for a quadratic algorithm.

O(2n) As the size of the problem increases, the time requirement for an

exponential algorithm increases too rapidly to be practical.

Fall 2006 CENG 707 20

Growth-Rate Functions

• If an algorithm takes 1 second to run with the problem size 8, what is the time requirement (approximately) for that algorithm with the problem size 16?

• If its order is:

O(1) T(n) = 1 second

O(log2n) T(n) = (1*log216) / log28 = 4/3 seconds

O(n) T(n) = (1*16) / 8 = 2 seconds

O(n*log2n) T(n) = (1*16*log216) / 8*log28 = 8/3 seconds

O(n2) T(n) = (1*162) / 82 = 4 seconds

O(n3) T(n) = (1*163) / 83 = 8 seconds

O(2n) T(n) = (1*216) / 28 = 28 seconds = 256 seconds

Fall 2006 CENG 707 21

Properties of Growth-Rate Functions

1. We can ignore low-order terms in an algorithm’s growth-rate function.– If an algorithm is O(n3+4n2+3n), it is also O(n3).

– We only use the higher-order term as algorithm’s growth-rate function.

2. We can ignore a multiplicative constant in the higher-order term of an algorithm’s growth-rate function.– If an algorithm is O(5n3), it is also O(n3).

3. O(f(n)) + O(g(n)) = O(f(n)+g(n))– We can combine growth-rate functions.

– If an algorithm is O(n3) + O(4n), it is also O(n3 +4n2) So, it is O(n3).

– Similar rules hold for multiplication.

Fall 2006 CENG 707 22

Some Mathematical Facts

• Some mathematical equalities are:

22

)1(*...21

2

1

nnnni

n

i

36

)12(*)1(*...41

3

1

22 nnnnni

n

i

122...21021

0

1

n

i

nni

Fall 2006 CENG 707 23

Growth-Rate Functions – Example1

Cost Timesi = 1; c1 1sum = 0; c2 1while (i <= n) { c3 n+1

i = i + 1; c4 nsum = sum + i; c5 n

}

T(n) = c1 + c2 + (n+1)*c3 + n*c4 + n*c5 = (c3+c4+c5)*n + (c1+c2+c3)= a*n + b

So, the growth-rate function for this algorithm is O(n)

Fall 2006 CENG 707 24

Growth-Rate Functions – Example2

Cost Timesi=1; c1 1sum = 0; c2 1while (i <= n) { c3 n+1

j=1; c4 nwhile (j <= n) { c5 n*(n+1) sum = sum + i; c6 n*n j = j + 1; c7 n*n

} i = i +1; c8 n}

T(n) = c1 + c2 + (n+1)*c3 + n*c4 + n*(n+1)*c5+n*n*c6+n*n*c7+n*c8= (c5+c6+c7)*n2 + (c3+c4+c5+c8)*n + (c1+c2+c3)= a*n2 + b*n + c

So, the growth-rate function for this algorithm is O(n2)

Fall 2006 CENG 707 25

Growth-Rate Functions – Example3

Cost Timesfor (i=1; i<=n; i++) c1 n+1

for (j=1; j<=i; j++) c2

for (k=1; k<=j; k++) c3

x=x+1; c4

T(n) = c1*(n+1) + c2*( ) + c3* ( ) + c4*( )

= a*n3 + b*n2 + c*n + d So, the growth-rate function for this algorithm is O(n3)

n

j

j1

)1(

n

j

j

k

k1 1

)1(

n

j

j

k

k1 1

n

j

j1

)1(

n

j

j

k

k1 1

)1(

n

j

j

k

k1 1

Fall 2006 CENG 707 26

What to Analyze• An algorithm can require different times to solve different

problems of the same size.– Eg. Searching an item in a list of n elements using sequential search. Cost:

1,2,...,n

• Worst-Case Analysis –The maximum amount of time that an algorithm require to solve a problem of size n.

– This gives an upper bound for the time complexity of an algorithm.– Normally, we try to find worst-case behavior of an algorithm.

• Best-Case Analysis –The minimum amount of time that an algorithm require to solve a problem of size n.

– The best case behavior of an algorithm is NOT so useful.

• Average-Case Analysis –The average amount of time that an algorithm require to solve a problem of size n.

– Sometimes, it is difficult to find the average-case behavior of an algorithm.– We have to look at all possible data organizations of a given size n, and their

distribution probabilities of these organizations.– Worst-case analysis is more common than average-case analysis.

Fall 2006 CENG 707 27

What is Important?

• We have to weigh the trade-offs between an algorithm’s time requirement and its memory requirements.

• We have to compare algorithms for both style and efficiency.– The analysis should focus on gross differences in efficiency and not reward coding

tricks that save small amount of time.

– That is, there is no need for coding tricks if the gain is not too much.

– Easily understandable program is also important.

• Order-of-magnitude analysis focuses on large problems.

Fall 2006 CENG 707 28

Sequential Searchint sequentialSearch(int a[], int x, int n){

int i; for (i = 0; i < n && a[i]!= x; i++);

if (i == n) return –1;

return i;}Unsuccessful Search: O(n)

Successful Search:Best-Case: item is in the first location of the array O(1)Worst-Case: item is in the last location of the array O(n)Average-Case: The number of key comparisons 1, 2, ..., n

O(n)n

nn

n

in

i 2/)( 21

Fall 2006 CENG 707 29

Binary Search

• Array must be sorted.

• at each step we reduce the length of the array to be searched by half.

• (e.g. array size = 1 million => 20 comparisons needed to locate any item)

… …

low mid high

Fall 2006 CENG 707 30

Binary Search Algorithm

int binarySearch(int a[], int size, int x){ int low =0; int high = size – 1; int mid; // mid will be the index of //target when it’s found. while (low <= high) { mid = (low + high)/2;

if (a[mid] < x) low = mid + 1;

else if (a[mid] > x) high = mid – 1;

else return mid;

} return –1;}

Fall 2006 CENG 707 31

Binary Search – Analysis

• For an unsuccessful search:

– The number of iterations in the loop is log2n + 1

O(log2n)

• For a successful search:

– Best-Case: The number of iterations is 1. O(1)

– Worst-Case: The number of iterations is log2n +1 O(log2n)

– Average-Case: The avg. # of iterations < log2n O(log2n)

0 1 2 3 4 5 6 7 an array with size 8

3 2 3 1 3 2 3 4 # of iterations

The average # of iterations = 21/8 < log28

Fall 2006 CENG 707 32

How much better is O(log2n)?

n O(log2n)

16 4

64 6

256 8

1024 (1KB) 10

16,384 14

131,072 17

262,144 18

524,288 19

1,048,576 (1MB) 20

1,073,741,824 (1GB) 30